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rational function - Rational function extra problems 3x3...

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Rational function extra problems: 8. Z 3 x 3 + 18 x 2 + 33 x - 2 ( x 2 + 6 x + 10) 2 dx Begin using integration by parts: 3 x 3 + 18 x 2 + 33 x - 2 ( x 2 + 6 x + 10) 2 = Ax + B x 2 + 6 x + 10 + Cx + D ( x 2 + 6 x + 10) 2 3 x 3 + 18 x 2 + 33 x - 2 = ( Ax + B )( x 2 + 6 x + 10) + Cx + D 3 x 3 + 18 x 2 + 33 x - 2 = ( A ) x 3 + (6 A + B ) x 2 + (10 A + B + C ) x + (10 B + D ) So we have a system of equations: A=3 6A+B=18 10A+6B+C=33 10B+D=-2 So we have that A=3, B=0, C=3, D=-2 Thus Z 3 x 3 + 18 x 2 + 33 x - 2 ( x 2 + 6 x + 10) 2 dx = Z 3 x x 2 + 6 x + 10 + 3 x - 2 ( x 2 + 6 x + 10) 2 dx Next we can complete the square to see that x 2 +6 x +10 = ( x +3) 2 +1 and use u substitution where u=x+3,x=u-3, du=dx So we get 3 Z u - 3 u 2 + 1 du + Z 3 u - 11 ( u 2 + 1) 2 du = 3 Z u u 2 + 1 du - 3 Z 1 u 2 + 1 du + 3 Z u ( u 2 + 1) 2 du - 11 Z 1 ( u 2 + 1) 2 du we can solve several of these with simple substitution v = u 2 + 1 , dv = 2 udu 1
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some we need a trig substitution of u = tanθ, du = sec 2 θdθ 3 Z u u 2 + 1 du - 3 Z 1 u 2 + 1 du + 3 Z u ( u 2 + 1) 2 du - 11 Z 1 ( u 2 + 1) 2 du = 3 2 Z 1 v dv - 3 Z 1 u 2 + 1 du + 3 2 Z 1 v 2 dv - 11 Z sec 2 θ (tan 2 θ + 1) 2 = 3 2 ln | v | - 3 arctan( u ) - 3 2 1 v - 11 Z sec 2 θ sec 4 θ = 3 2 ln | u 2 + 1 | - 3 arctan( u ) - 3 2 1 ( u 2
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